On Coupled Klein-Gordon-Schrödinger Equations with Acoustic Boundary Conditions

We are concerned with the existence and energy decay of solution to the initial boundary value problem for the coupled Klein-Gordon-Schrödinger equations with acoustic boundary conditions.

In this paper, we are concerned with global existence and uniform decay for the energy of solutions of Klein-Gordon-Schrödinger equations:

where is a bounded domain, , with boundary of class , where and are two disjoint pieces of each having nonempty interior and are given functions. We will denote by the unit outward normal vector to . stands for the Laplacian with respect to the spatial variables; denotes the derivative with respect to time . Here is the normal displacement to the boundary at time with the boundary point .

The above equations describe a generalization of the classical model of the Yukawa interaction of conserved complex nucleon field with neutral real meson field. Here, is complex scalar nucleon field while and are real scalar meson one.

In three dimension, [1–5] studied the global existence for the Cauchy problem to

Klein-Gordon-Schrödinger equations have been studied as many as ever by many authors (cf. [6–11], and a list of references therein). However, they did not have treated acoustic boundary conditions.

Boundary conditions of the fifth and sixth equations are called acoustic boundary conditions. Equation (1.1)₅ (the fifth equation of (1.1)) does not contain the second derivative , which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as in (1.1)₆ (the sixth equation of (1.1)) is concerned to the case of a nonporous boundary, (1.1)₆ simulates a porous boundary when a function is nonnegative. When general acoustic boundary conditions, which had the presence of in (1.1)₅, are prescribed on the whole boundary, Beale [12–14] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semigroup methods. The asymptotic behavior was obtained in [13], but no decay rate was given there. Recently, the acoustic boundary conditions have been treated by many authors (cf. [15–21] and a list of references therein). However, energy decay problem with acoustic boundary conditions was studied by a few authors. For instance, Rivera and Qin [22] proved the polynomial decay for the energy of the wave motion using the Lyapunov functional technique in the case of general acoustic boundary conditions and . Frota and Larkin [23] considered global solvability and the exponential decay of the energy for the wave equation with acoustic boundary conditions, which eliminated the second derivative term for . However, it is not simple to apply the semigroup theory as well as Galerkin's method in [23] because a system of corresponding ordinary equations is not normal and one cannot apply directly the Carathéodory's theorem. So they overcame this problem using the degenerated second order equation. And Park and Ha [20] studied the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-type equation by using the multiplier technique. Moreover, [20] proved the exponential and polynomial decay rates of solutions for all .

In this paper, we prove the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-Schrödinger equations with acoustic boundary conditions and allow to apply the method developed in [23]. However, [23] did not treat the Klein-Gordon-Schrödinger equations.

This paper is organized as follows. In Section 2, we recall the notation and hypotheses and introduce our main result. In Section 3, using Galerkin's method, we prove the existence and uniqueness of solutions to problem (1.1). In Section 4, we prove the exponential energy decay rate for the solutions obtained in Section 3.

We begin this section introducing some notations and our main results. Throughout this paper we define the Hilbert space with the norm , where is an -norm and ; without loss of generality we denote . Moreover, -norm and -norm are denoted by and , respectively. Denoting by and the trace map of order zero and the Neumann trace map on , respectively, we have

and the generalized Green formula

holds for all and . We denoted . By the Poincare's inequality, the norm is equivalent to the usual norm from . Now we give the hypotheses for the main results.

Hypotheses on

Let be a bounded domain in , with boundary of class . Here and are two disjoint pieces of , each having nonempty interior and satisfying the following conditions:

where represents the unit outward normal vector to .

Hypotheses on , , and

Assume that are continuous functions such that

where

Moreover, , , , and is a real constant.

In physical situation, and are parameters representing the gratitude of diffusion and dissipation effects. Also, is a fluid density and describes the mass of a meson. Boundary condition (1.1)₆ (the sixth equation of (1.1)) simulates a porous boundary because of the function .

We define the energy of system (1.1) by

Now, we are in a position to state our main result.

Theorem 2.1.

Let satisfy the inequality

where is a positive constant. Assume that and hold. Then problem (1.1) has a unique strong solution verifying

Moreover, if satisfies

then one has the following energy decay:

where and are positive constants.

Note

By the hypothesis in , we have for all , so we can assume (2.9).

In this section, we prove the existence and uniqueness of solutions to problem (1.1). Let and be orthonormal bases of and , respectively, and define and . Let , , and be sequences of such that , strongly in , and strongly in . For each and , we consider

satisfying the approximate perturbed equations

where and for all , , . The local existence of regular functions , , and is standard, because (3.2) is a normal system of ordinary differential equation. A solution to the problem (1.1) on some interval will be obtained as the limit of as and . Then, this solution can be extended to the whole interval , for all , as a consequence of the a priori estimates that will be proved in the next step.

3.1. The First Estimate

Replacing , , and by , , and in (3.2), respectively, we obtain

Taking the real part in (3.3), we get

On the other hand, by Young's inequality we have

Substituting the above inequality in (3.6), and then integrating (3.6) over with , we get

Using the fact that , , (2.7) and Gronwall's lemma, we obtain

where is a positive constant which is independent of , , and .

3.2. The Second Estimate

First of all, we are going to estimate . By taking in (3.2)₃ (the third equation of (3.2)), we get

By considering and hypotheses on the initial data, for all and , we obtain

Now, by replacing and by and in (3.2), respectively, also differentiating (3.2)₃ with respect to , and then substituting , we have

We now estimate the last term on the left-hand side of (3.12) and the term on the right-hand side of (3.12). Applying Green's formula, we deduce

Considering the equality

for all , we have

Hence,

Also,

Hence,

Replacing the above calculations in (3.12) and then taking the real part, we obtain

On the other hand, we can easily check that

Therefore,

Replacing (3.23) in (3.13) and using the imbedding , we have

where is an imbedding constant.

Adding (3.14), (3.21), and (3.24), we get

By choosing and integrating (3.25) from to , we have

where is a positive constant. Using the hypotheses on and , (2.7), (3.11), and Gronwall's lemma, we obtain

where is a positive constant which is independent of , , and .

3.3. The Third Estimate

First of all, we are going to estimate and . By taking in (3.2), we get

By considering and and hypotheses on the initial data, for all and , we obtain

where and are positive constants.

Now by differentiating (3.2) with respect to and substituting , , and , we have

Taking the real part in (3.31), we infer

Considering the equality

for all , we have

Also,

From (3.34)–(3.37), we conclude

On the other hand, we can easily check that

Therefore,

Replacing (3.40) in (3.32), we have

Combining (3.33), (3.38), and (3.41), we obtain

Integrating (3.42) from to , we have

Therefore, using the hypotheses on and , (2.7), (3.11), (3.29), (3.30), and Gronwall's lemma, we get

where is a positive constant which is independent of , , and .

According to (3.9), (3.27), and (3.44), we obtain that

From (3.45)–(3.52), there exist subsequences , , and , which we still denote by , and , respectively, such that

We can see that (3.9), (3.27), and (3.44) are also independent of . Therefore, by the same argument as (3.45)–(3.61) used to obtain , , and from , , and , respectively, we can pass to the limit when in , , and , obtaining functions , , and such that

Thus, by the above convergences and (3.53), we can prove the existence of solutions to (1.1) satisfying (2.8).

3.4. Uniqueness

Let and be two-solution pair to problem (1.1). Then we put

From (3.2), we have

for all and . By replacing , , and in (3.64), it holds that

Taking the real part in (3.65), we get

We now estimate the last term on the left-hand side of (3.68) and the term on the right-hand side of (3.68). We can easily check that

By using the fact that for all , we obtain

Also,

Hence by Hölder's inequality, (3.45), and (3.47), we deduce

where is a positive constant. Replacing (3.70) and (3.72) in (3.68), we have

On the other hand, we can easily check that

where is a positive constant. Therefore, we can rewrite (3.66) as

Adding (3.67), (3.73), and (3.75), we get

Applying Gronwall's lemma, we conclude that . This completes the proof of existence and uniqueness of solutions for problem (1.1).

Multiplying the first equation of (1.1) by and integrating over , we get

Taking the real part in the above equality, it follows that

Now, multiplying the second equation of (1.1) by and integrating over , we have

Taking into account (1.1)₅ and (1.1)₆ (the fifth and sixth equations of (1.1)), we can see that

Therefore (4.3) can be rewritten as

Adding (4.2) and (4.5), we obtain

By choosing and the hypotheses on , we get

So we conclude that is a nonincreasing function.

Now we consider a perturbation of . For each , we define

where

By definition of the function , Poincare's inequality, and imbedding theorem, we have

where is a Poincare constant. Hence, from (4.8) and (4.10), there exists a positive constant such that

for all and . This means that there exist positive constants and such that

On the other hand, differentiating , we have

where

Now, we will estimate the terms on the right-hand side of (4.14).

Estimates for

Using the first equation of (1.1), we can easily check that

Estimates for

Applying Green's formula, we deduce

Estimates for

Using the second equation of (1.1) and Young's inequality, we have

Estimates for

Similar to estimates for we have

By replacing (4.15)–(4.18) in (4.14) and choosing and , we conclude that

From (2.9) and (4.19), we obtain

We now estimate the last term on the right-hand side of (4.20).

Estimates for

From the fifth equation of (1.1), we have

Estimates for

By Young's inequality, we have

where is an arbitrary positive constant. By replacing (4.21) and (4.22) in (4.20), we get

We note that

where . By the above inequality and choosing such that

we conclude that

where is a positive constant. Now choosing sufficiently small, we obtain

where is a positive constant. Therefore,

From (4.12), we have

This implies the proof of Theorem 2.1 is completed.

1. Bachelot, A: Problème de Cauchy pour des systèmes hyperboliques semilinéaires. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire. 1(6), 453–478 (1984)

2. Baillon, J-B, Chadam, JM: The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations. In: Medeiros (ed.) Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Jane, North-Holland Mathematics Studies, vol. 30, pp. 37–44. North-Holland, Amsterdam, The Netherlands (1978)

3. Fukuda, I, Tsutsumi, M: On coupled Klein-Gordon-Schrödinger equations. I. Bulletin of Science and Engineering Research Laboratory. Waseda University.(69), 51–62 (1975)

4. Fukuda, I, Tsutsumi, M: On coupled Klein-Gordon-Schrödinger equations. II. Journal of Mathematical Analysis and Applications. 66(2), 358–378 (1978). Publisher Full Text

5. Fukuda, I, Tsutsumi, M: On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions. Proceedings of the Japan Academy. 51(6), 402–405 (1975). Publisher Full Text

6. Bisognin, V, Cavalcanti, MM, Cavalcanti, VND, Soriano, J: Uniform decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Nonlinear Differential Equations and Applications. 15(1-2), 91–113 (2008). Publisher Full Text

7. Cavalcanti, MM, Cavalcanti, VND: Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations. Nonlinear Differential Equations and Applications. 7(3), 285–307 (2000). Publisher Full Text

8. Colliander, J, Holmer, J, Tzirakis, N: Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems. Transactions of the American Mathematical Society. 360(9), 4619–4638 (2008). Publisher Full Text

9. Miao, C, Xu, G: Global solutions of the Klein-Gordon-Schrödinger system with rough data in . Journal of Differential Equations. 227(2), 365–405 (2006). Publisher Full Text

10. Park, JY, Kim, JA: Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations with non-linear boundary damping and memory term. Mathematical Methods in the Applied Sciences. 29(9), 947–964 (2006). Publisher Full Text

11. Tzirakis, N: The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space. Communications in Partial Differential Equations. 30(4–6), 605–641 (2005)

12. Beale, JT: Spectral properties of an acoustic boundary condition. Indiana University Mathematics Journal. 25(9), 895–917 (1976). Publisher Full Text

13. Beale, JT: Acoustic scattering from locally reacting surfaces. Indiana University Mathematics Journal. 26(2), 199–222 (1977). Publisher Full Text

14. Beale, JT, Rosencrans, SI: Acoustic boundary conditions. Bulletin of the American Mathematical Society. 80, 1276–1278 (1974). Publisher Full Text

15. Cousin, AT, Frota, CL, Larkin, NA: Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions. Funkcialaj Ekvacioj. 44(3), 471–485 (2001)

16. Cousin, AT, Frota, CL, Larkin, NA: On a system of Klein-Gordon type equations with acoustic boundary conditions. Journal of Mathematical Analysis and Applications. 293(1), 293–309 (2004). Publisher Full Text

17. Frota, CL, Goldstein, JA: Some nonlinear wave equations with acoustic boundary conditions. Journal of Differential Equations. 164(1), 92–109 (2000). Publisher Full Text

18. Ha, TG, Park, JY: Existence of solutions for the Kirchhoff-type wave equation with memory term and acoustic boundary conditions. Numerical Functional Analysis and Optimization. 31(8), 921–935 (2010). Publisher Full Text

19. Mugnolo, D: Abstract wave equations with acoustic boundary conditions. Mathematische Nachrichten. 279(3), 299–318 (2006). Publisher Full Text

20. Park, JY, Ha, TG: Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. Journal of Mathematical Physics. 50(1), (2009)

21. Vicente, A: Wave equation with acoustic/memory boundary conditions. Boletim da Sociedade Paranaense de Matemática. 27(1), 29–39 (2009)

22. Muñoz Rivera, JE, Qin, Y: Polynomial decay for the energy with an acoustic boundary condition. Applied Mathematics Letters. 16(2), 249–256 (2003). Publisher Full Text

23. Frota, CL, Larkin, NA: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Contributions to Nonlinear Analysis, Progress in Nonlinear Differential Equations and Their Applications, pp. 297–312. Birkhäuser, Basel, Switzerland (2006).